This thesis explores the application of Kalman filtering techniques to enhance pairs trading strategies in financial markets. Pairs trading is a statistical arbitrage strategy that exploits temporary price divergences between historically correlated assets by taking opposite positions with the expectation of mean reversion. The study addresses a fundamental challenge in pairs trading: accurately modeling the underlying spread dynamics in the presence of market noise. The research implements a state-space model framework where the observed spread between asset prices is treated as a noisy measurement of a mean-reverting process. A default Kalman filter is applied to estimate the true underlying spread by filtering out market noise, with the goal of generating more reliable trading signals. To optimize the Kalman filter’s performance, the ExpectationMaximization (EM) algorithm is employed to estimate the model’s latent parameters, including process noise and observation noise covariances. The methodology is tested on a cryptocurrency pair (Ethereum-NEO) identified from existing literature using the distance method for pair selection, covering the period from January 2018 to December 2019. To test the performance of Kalman filtering three approaches are compared: trading on unfiltered spreads, trading on Kalman-filtered spreads with default parameters, and trading on spreads filtered using EM-optimized parameters. The empirical results reveal several key findings. Surprisingly, the unfiltered spread strategy initially outperformed the default Kalman filter approach, generating $725.73 in profits across 4 trades compared to $523.95 across 3 trades for the filtered approach. However, when EM optimization was applied, the Kalman filter strategy achieved the highest performance with $750.83 in profits across 4 trades. A notable discovery is that the estimated state transition coefficient consistently converged to 1, indicating random walk behavior rather than the expected mean-reverting dynamics. This suggests that the theoretical assumption of mean reversion may not always align with empirical data, highlighting the importance of model validation in quantitative finance applications. The study demonstrates that while Kalman filtering can enhance pairs trading strategies, parameter optimization through EM is crucial for achieving superior performance. The research contributes to the understanding of noise reduction techniques in financial time series and provides insights into the practical challenges of implementing statistical arbitrage strategies. Future work could explore larger asset universes, explicit mean-reversion constraints, and the incorporation of transaction costs and risk management considerations. ... Read more

In many situations, when we have a group of people, they all formopinions on a subject. Everyone in a population influences each others opinion. Naturally, people how the opinion of children are influenced by other children differs from how adults influence their opinions. Between different age groups there are all kind of different interactions.
This thesis aims to model how such opinions evolve and influence each other over time. First We assume an individual can either have a negative or positive opinion on a subject. To model this, we use the Ising model, originally developed for the description of magnetism in metals. We model opinion changes as random processes influenced by the opinion of other individuals in the population. We divide the population into subgroups of people who interact similarly. In this thesis, we prove that if we make the total group of people larger and larger, this random process becomes a deterministic process. Just like when you flip a coin infinitely many times, you end up with heads 50% of the time. We then determine how the different populations influence each other’s opinions. Understanding group opinion dynamics can help explain the spread of misinformation on social media, the emergence and disappearance of political parties, or how companies can predict or start trends. ... Read more

Variational inference comprises a family of statistical methods to obtain the optimal approximation of a target probability distribution using some reference class of distributions and a cost function, commonly the Kullback-Leibler (KL) divergence. Recent work on variational inference has yielded a fast, stable set of mean and covariance evolutions which dynamically yield variational Gaussian approximations via a restriction to Gaussian measures of the well-known JKO scheme. The sequence of Gaussian measures thus generated converges towards the KL-optimal Gaussian approximation of the VI target: it may also be used to approximate the entire sequence of distributions generated by a JKO gradient flow directed at this same target, thereby supporting practical usage of Gaussian VI as well as fast, approximate modelling of the Fokker-Planck PDE. However, it is not immediately clear whether this Gaussian sequence offers valid, helpful approximations of the original JKO gradient flow. In this work, three upper bounds for the sequence of Wasserstein-2 distances between the two gradient flows are obtained by exploiting the Riemannian structure of the W2 manifold and the shared properties of the Gaussian and JKO evolutions. Numerical simulations support the validity of these bounds and test their performance in both ordinary and exceptional scenarios. One of the bounds may be computed solely using the Gaussian evolution and the target potential, thus offering a tractable estimator for the suitability of variational Gaussian approximations which retains the attractive properties of Wasserstein distances whilst avoiding their computational demands. ... Read more

This thesis focuses on two main topics: large deviations for Markov processes and the well-posedness of Hamilton–Jacobi equations. The first two chapters provide an introduction to both areas. Chapter 1 explores the mathematical foundations of Hamilton–Jacobi equations, highlighting their applications in control theory and emphasising the role of viscosity solutions in handling situations where classical solutions fail. Chapter 2 introduces large deviations theory, starting from basic examples and leading to rigorous definitions. A key theme is the connection between large deviations and Hamilton–Jacobi equations, introduced through the Feng–Kurtz method. The subsequent chapters present the main research contributions of this thesis. Chapter 3 studies two examples of two-scale Markov processes and applies the Feng–Kurtz method to establish a large deviations principle. Chapter 4 transitions to the second theme of this thesis: the well-posedness of Hamil- ton–Jacobi equations. Motivated by the previous examples, we analyse a general class of Hamilton–Jacobi equations. We establish a comparison principle for viscosity solutions, demonstrating its applicability in a broad setting. Chapter 5 extends these results by proving the existence of viscosity solutions for a general class of Hamilton–Jacobi equations using Lyapunov control techniques. In the final chapter, Chapter 6, we investigate second-order Hamilton–Jacobi equations, presenting a novel proof of the comparison principle for viscosity solutions. ... Read more

In this thesis, we study large deviations and parameter estimations for small noise diffusion processes. In Chapter 1, we start with the classical limit theorems to intuitively introduce large deviations and parameter estimations, which provide for further developments in the thesis.

The first part, consisting of Chapters 2 - 4, is on large deviations. In Chapter 2, we begin with the simple stochastic differential equation to explain the idea behind the proof of the nonlinear semigroup method, which is used to prove large deviations in Chapters 3 and 4. In the process, viscosity solutions and the Hamilton-Jacobi-Bellman equations are introduced... ... Read more

This thesis focusses on the study of multi-layer particle systems with an emphasis on the scaling limits of particle systems of this type. Chapter 1 gives an overall introduction to the field of statistical physics and interacting particle systems, and gives a motivation on the study of multi-layer particle systems. In Chapter 2 we introduce the mathematical background required for this thesis in a rigorous manner. The topics discussed include Markov semigroups and generators, path-space convergence, ergodic theory, martingales, couplings, and duality. In Chapter 3 we introduce three types of multi-layer particle systems; the multi-layer exclusion process, the multi-layer inclusion process and the run-and-tumble particle process. We then characterize the ergodic measures with a finite moment condition for these three processes using duality and successful couplings. In Chapter 4 we study the hydrodynamic limit and the stationary fluctuations of the multi-layer run-and-tumble particle process, and use them to infer the same scaling limits for the total density. Furthermore, by an application of Schilder's theorem, we find a large deviation result for the fluctuation field of the total density. In Chapter 5 we establish a large deviation principle for the multi-species stirring process. The method of proof involves studying the hydrodynamic limit of a weakly asymmetric process and a superexponential estimate. In Chapter 6 we return to the multi-layer setting and establish a large deviation principle for the run-and-tumble particle process on two layers with an added mean-field interaction, meaning that the switching between the layers depends on the magnetization of the process. We end with a first step towards an explicit large deviation principle of the total density. ... Read more

Estimating the state of dynamically evolving systems is a fundamental challenge across diverse fields such as robotics, navigation, economics, and environmental monitoring. This thesis explores and compares three prominent state estimation methods: the Kalman Filter (KF), the Extended Kalman Filter (EKF), and the Unscented Kalman Filter (UKF), each tailored to handle specific complexities encountered in real-world applications.

The foundational Kalman Filter is rigorously examined first, deriving its algorithm through Bayesian inference and the fusion of multiple estimates. A comparative analysis of these approaches highlights the KF’s robustness in linear systems while acknowledging limitations in nonlinear environments.

The thesis then transitions to the Extended Kalman Filter, which extends the KF to nonlinear systems by linearizing state equations. Detailed mathematical derivation and comparative studies underscore the EKF’s enhanced capabilities in handling complex dynamics, yet reveal challenges in accuracy and computational cost.

Moving further, the Unscented Kalman Filter is introduced as a non-linear state estimation method utilizing the Unscented Transform. Detailed exploration and mathematical formulation demonstrate its effectiveness in addressing uncertainties, presenting a viable alternative to both KF and EKF in scenarios where linearization proves inadequate.

To validate these methodologies, simulations are conducted using real-world data from the KITTI dataset, comprising of GPS and IMU measurements. Ground truth trajectories and non-linear variables such as yaw rates and forward velocities are utilized, showcasing each filter’s ability to estimate and track dynamic system states accurately.

Results from simulations are analyzed using performance metrics including Normalized Estimation Error Squared (NEES) and Root Mean Squared Error (RMSE), providing quantitative insights into filter performance relative to ground truth. These evaluations emphasize the strengths and limitations of each method across various application domains, supporting informed decisions on filter selection based on specific system dynamics and measurement characteristics.

In conclusion, this thesis contributes to a comprehensive analysis and comparative study of state estimation methods essential for navigating the complexities of dynamically evolving systems. By bridging theoretical advancements with practical insights, it lays a foundation for future research and application in fields requiring precise state estimation amidst dynamic change. ... Read more

The Kalman filter is a recursive algorithm that estimates the state of a dynamic system subject to measurement and model noise. If all noise terms affecting the system are white Gaussian noise with known mean and variance, and all noise terms are independent of each other, then the Kalman filter is the optimal estimator for the state variable. When measurements are collected from multiple sources, the covariance between these sources should be known or the sources should be independent to ensure that the estimate made by the Kalman filter is optimal. When the covariance between dependent measurement sources is not known, various methods exist which provide a solution to this problem. This thesis discusses two methods: the H∞ filter and covariance intersection... ... Read more

The Curie-Weiss model is a simplification of the Ising model to show the existence of a phase transition for ferromagnetism. In this thesis, we study the behaviour of sums of these dependent variables. We prove in general that under the appropriate assumptions, we can still conclude a version of the Law of Large Numbers. We also find that if there exists a certain m∈R, λ>0 and integer k≥1, we have that (S_n-nm)/n^1/2k converges to exp(-λs^2k/(2k)!) in distribution.
For the Curie-Weiss model this means that for β, which is a constant proportion to inverse temperature, we find that if β∈(0,1) we have S_n/n→δ(s) and S_n/√n→ N(0,σ²) in distribution where σ²=(1-β)^-1-1. At β=1 there occurs a phase transition, we still have that S_n/n→δ(s), but now S_n/n^3/4→\exp(-s⁴/12). When β>1 we can find an m>0 such that S_n/n→½[δ(s-m)+δ(s+m)].
We also study the Curie-Weiss model where we assume that it is under the influence of a magnetic field. We prove that we do not find a phase transition, and we always have S_n/n→δ(s-m) in distribution for some m∈R. Next to this we find that (S_n-nm)/√n always converges to a normal distribution. ... Read more

In this bachelor thesis we use a stochastic model to aspire to explain biodiversity patterns in different ecosystems with selection advantage. The stochastic model we use is an extension of the mean­-field voter model where we include a selection factor. In the model individuals with two different types of alleles in two different ecosystems are considered. The model is a stochastic Markov process that describes interactions of individuals with each­other over time. This means that the ratio of individuals with certain alleles stochastically drifts over time. The main goal of this bachelor thesis is investigate whether is it possible that individuals with two different types of alleles can coexist (a stable equi­librium) in two populations. We do this by taking the limit of this Markov process such that we can show convergence to ordinary differential equations. By studying these differential equations we ob­tain results: vector fields with equilibrium points. We conclude that, under certain conditions and with a selection advantage, coexistence of individuals with two different alleles in two different ecosystems is possible (see Section 4.4) in the form of a stable equilibrium. Furthermore we claim that the typical time to absorption, reaching an absorbing state where all the individuals have the same allele, of the two­-dimensional mean­-field voter model with selection scales exponentially with the system size. ... Read more

We inspect the behavior of the probability that a weighted sum of random variables with log-normal tails is greater than its expected value. Under the right conditions for the weights and the variance being set to 1; we were able to bound a suitable transformation of this probability with the upper bound being a fixed factor of the square root of e above the lower bound. Beyond this, we analyse the conditions on the weights and determine a method for letting the weights be random and give an example.
We end off by extending our result to general variance, where we see that the deviation between the lower and upper bound as well as the domain for the result are dependant on the variance. ... Read more

As an insurer you want identify the risks you take to prevent bankruptcy. The
theory of large deviations formalizes the study of such rare events. We will use the
theorem of Cramér, which is a main theorem in large deviation theory, to investigate
the rate at which the probability of large deviations of the sums of random variables
decay. Using Sanov’s theorem we will derive an expression for large deviations of
the empirical measure. Furthermore, we will use Gibbs’s principle to derive the
distribution of random variables conditional on a large deviation.
... Read more